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When working with integers, specifically in multiplicative operations, it's important to remember a few key rules. We determine the sign of the result based on the signs of the numbers being multiplied. For two integers:

• If both numbers have the same sign, their product is positive.
• If the numbers have different signs, their product is negative.

Using these rules helps ensure that our multiplications are accurate. Multiplying several numbers together can be simplified step-by-step. Let鈥檚 apply these principles to our example problem:
First, we have: \( 3 \times (-10) \)
Since we are multiplying a positive by a negative number, our result will be negative: \(-30\).
Next, we have \(-30 \times (-1)\): Two negatives make a positive, so we get \(30\).
Finally, we multiply \(30 \times (-2)\): A positive times a negative equals a negative, so the result is \(-60\). Understanding these rules will make multiplication of integers much easier for you!

Negative numbers are numbers less than zero. They are represented with a minus sign (-). When dealing with negative numbers in multiplication, it is crucial to remember how they interact with each other:

• Multiplying a negative by a positive always results in a negative product.
• Multiplying a negative by another negative results in a positive product.

A common mistake is to forget the rule that 鈥渢wo negatives make a positive.鈥� Keeping the properties of negative numbers in mind can help avoid mistakes. For instance, in our example, during the second step where \(-30 \times (-1)\), it converts to a positive \(30\). This fact is the foundation to correctly solving the problem and similar exercises.

Simplification is the process of reducing an expression to its most basic form. When simplifying a multiplication problem with several integers of different signs, it鈥檚 helpful to take it step-by-step:
1. Simplify the expression by multiplying two numbers at a time.
2. Apply the integer multiplication rules:
- Positive 脳 Positive = Positive
- Negative 脳 Positive = Negative
- Negative 脳 Negative = Positive
3. Continue to simplify until you reach the final product. For example, with the initial expression \(3 \times (-10) \times (-1) \times (-2)\), start by handling two numbers at a time. This approach avoids overwhelming complexity and ensures you're constantly applying the correct rules.
By simplifying step-by-step, as shown in each stage of our solution, you can more easily manage and verify each part of the problem. Practicing simplification like this can improve accuracy and confidence in solving such problems.